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Convergence Analysis of Piecewise Polynomial Collocation Methods for System of Weakly Singular Volterra Integral Equations of The First Kind

Received: 21 March 2017     Accepted: 22 March 2017     Published: 11 April 2017
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Abstract

We study regularity of solutions of weakly singular Volterra integral equations of the first kind. We then study the numerical analysis of discontinuous piecewise polynomial collocation methods for solving such systems. The main purpose of this paper is the derivation of global convergent and super-convergent properties of introduced methods on the graded meshes. We apply relevant methods to a system of fractional differential equations and analyze them. The numerical experiments confirm the theoretical results.

Published in Applied and Computational Mathematics (Volume 7, Issue 1-1)

This article belongs to the Special Issue Singular Integral Equations and Fractional Differential Equations

DOI 10.11648/j.acm.s.2018070101.11
Page(s) 1-11
Creative Commons

This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.

Copyright

Copyright © The Author(s), 2017. Published by Science Publishing Group

Keywords

Discontinuous Piecewise Polynomial Spaces, Collocation Methods, Graded Meshes, Weakly Singular Volterra Integral Equations

References
[1] K. E. Atkinson, “The numerical solution of integral equations of the second kind,” Vol. 4, Cambridge university press, 1997.
[2] H. W. Branca, “The nonlinear Volterra equation of abel's kind and its numerical treatment,” Comput, 1978, 20, 307-324.
[3] H. Brunner, “Collocation methods for Volterra integral and related functional differential equations,” Vol. 15, Cambridge University Press, 2004.
[4] H. Brunner, A. Pedas and G. Vainikko, “Piecewise polynomial collocation methods for linear Volterra integro-differential equations with weakly singular kernels,” SIAM Journal on Numerical Analysis, 2001, 39, 957-982.
[5] R. Cameron and S. McKee, “Product integration methods for second-kind Abel integral equations,” Journal of Computational and Applied Mathematics, 1984, 11, 1-10.
[6] P. Eggermont, “A new analysis of the trapezoidal-discretization method for the numerical solution of Abel-type integral equations,” Journal of Integral Equations and Applications, 1981, 3, 317-332.
[7] R. Kress, V. Maz'ya and V. Kozlov, “Linear integral equations,” Vol. 82, Springer, 1989.
[8] C. Lubich, “Fractional linear multistep methods for Abel-Volterra integral equations of the first kind,” IMA Journal of Numerical Analysis, 1987, 7, 97-106.
[9] A. Pedas and E. Tamme, “On the convergence of spline collocation methods for solving fractional differential equations,” Journal of Computational and Applied Mathematics, 2011, 235, 3502-3514.
[10] H. Te Riele and P. Schroevers, “A comparative survey of numerical methods for the linear generalized Abel integral equation,” ZAMM- Journal of Applied Mathematics and Mechanics, 1986, 66, 163-173.
[11] A. Saadatmandi and M. Dehghan, “A collocation method for solving Abel’s integral equations of first and second kinds,” Zeitschrift für Naturforschung A, 2008, 63, 752-756.
[12] B. Shiri, “Numerical solution of higher index nonlinear integral algebraic equations of Hessenberg type using discontinuous collocation methods,” Mathematical Modelling and Analysis, 2014, 19, 99-117.
[13] V. Volterra, “Sulla inversione degli integrali definiti,” Atti della Accademia delle scienze di Torino,1896, 31, 311-323.
[14] R. Weiss, “Numerical procedures for volterra integral equations,” Bulletin of the Australian Mathematical Society, 1973, 8, 477-478.
[15] R. Weiss, “Product integration for the generalized Abel equation,” Mathematics of Computation, 1972, 26, 177-190.
[16] R. Weiss and R. Anderssen, “A product integration method for a class of singular first kind Volterra equations,” Numerical Mathematics, 1971, 18, 442-456.
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  • APA Style

    Gholamreza Karamali, Babak Shiri, Mahnaz Kashfi. (2017). Convergence Analysis of Piecewise Polynomial Collocation Methods for System of Weakly Singular Volterra Integral Equations of The First Kind. Applied and Computational Mathematics, 7(1-1), 1-11. https://doi.org/10.11648/j.acm.s.2018070101.11

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    ACS Style

    Gholamreza Karamali; Babak Shiri; Mahnaz Kashfi. Convergence Analysis of Piecewise Polynomial Collocation Methods for System of Weakly Singular Volterra Integral Equations of The First Kind. Appl. Comput. Math. 2017, 7(1-1), 1-11. doi: 10.11648/j.acm.s.2018070101.11

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    AMA Style

    Gholamreza Karamali, Babak Shiri, Mahnaz Kashfi. Convergence Analysis of Piecewise Polynomial Collocation Methods for System of Weakly Singular Volterra Integral Equations of The First Kind. Appl Comput Math. 2017;7(1-1):1-11. doi: 10.11648/j.acm.s.2018070101.11

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  • @article{10.11648/j.acm.s.2018070101.11,
      author = {Gholamreza Karamali and Babak Shiri and Mahnaz Kashfi},
      title = {Convergence Analysis of Piecewise Polynomial Collocation Methods for System of Weakly Singular Volterra Integral Equations of The First Kind},
      journal = {Applied and Computational Mathematics},
      volume = {7},
      number = {1-1},
      pages = {1-11},
      doi = {10.11648/j.acm.s.2018070101.11},
      url = {https://doi.org/10.11648/j.acm.s.2018070101.11},
      eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.s.2018070101.11},
      abstract = {We study regularity of solutions of weakly singular Volterra integral equations of the first kind. We then study the numerical analysis of discontinuous piecewise polynomial collocation methods for solving such systems. The main purpose of this paper is the derivation of global convergent and super-convergent properties of introduced methods on the graded meshes. We apply relevant methods to a system of fractional differential equations and analyze them. The numerical experiments confirm the theoretical results.},
     year = {2017}
    }
    

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    AU  - Gholamreza Karamali
    AU  - Babak Shiri
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    UR  - https://doi.org/10.11648/j.acm.s.2018070101.11
    AB  - We study regularity of solutions of weakly singular Volterra integral equations of the first kind. We then study the numerical analysis of discontinuous piecewise polynomial collocation methods for solving such systems. The main purpose of this paper is the derivation of global convergent and super-convergent properties of introduced methods on the graded meshes. We apply relevant methods to a system of fractional differential equations and analyze them. The numerical experiments confirm the theoretical results.
    VL  - 7
    IS  - 1-1
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Author Information
  • Shahid Sattari Aeronautical University of Science and Technology, South Mehrabad, Tehran, Iran

  • Shahid Sattari Aeronautical University of Science and Technology, South Mehrabad, Tehran, Iran

  • Department of Applied Mathematics, University of Tabriz, Bahman Boulevard, Tabriz, Iran

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